TITLE: Is there a "motivic Gromov-Witten invariant"?
QUESTION [17 upvotes]: I recently attended an interesting seminar, where the concept of motivic Donaldson-Thomas invariants was explained (0909.5088). 
Very roughly, the DT invariant is a generating function $\sum q^k e(M_k)$ of a numerical invariant $e(\cdot)$ of a sequence of moduli spaces $M_k$. The motivic DT invariant is obtained by considering $\sum q^k [M_k]$ where $[M_k]$ is the image in $K(Var)$.
This contains more info than the ordinary DT invariant.
Can this idea be applied for, say, the GW invariant of Calabi-Yau 3-folds, to get a finer invariant?
(Sorry for my vague question.)

REPLY [8 votes]: Let me just add a bit to what Balazs said. The fact that the moduli spaces of sheaves on a CY3 have this symmetric obstruction theory is a reflection of properties of the category they live in, namely the derived category of coherent sheaves on the CY3. Indeed, Kontsevich and Soibelman's general construction of motivic DT invariants applies to a general class of CY3 categories and the motivic invariants live in a Hall algebra associated to this category.
My point is that the natural category associated to Gromov-Witten invariants is the Fukaya category and so I would expect any sort of "motivic" version of GW theory to live in something associated to the Fukaya category. In particular, I don't think these invariants will live in the Grothendieck group of varieties the way the motivic DT invariants do. GW theory (A-model) is inherently analytic (or symplectic) whereas DT theory (B-model) is inherently algebraic. 
It would be really interesting to figure out what the analog of the Hall algebra is in the Fukaya category. Maybe the symplectic geometers already know it?